第7章単位円:正弦関数と余弦関数The Unit Circle: Sine and Cosine Functions

Introduction to The Unit Circle: Sine and Cosine Functions単位円への導入: 正弦関数と余弦関数

Two boats at a dock during low tide — The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr)潮は決まった、見通せる間隔で満ち引きする。(credit: Andrea Schaffer, Flickr)

• 7.1 Angles • 7.2 Right Triangle Trigonometry • 7.3 Unit Circle • 7.4 The Other Trigonometric Functions

・7.1 角・7.2 直角三角形の三角法・7.3 単位円・7.4 そのほかの三角関数

Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. And as a result of the motion of the moon itself, the tides occur with different strengths. Throughout history, many Indigenous peoples have used this regularity to build cultural narratives and direct key activities, such as agriculture, hunting, and fishing. Aboriginal people in the Torres Strait area (the northern tip) of Australia used the tidal peaks to determine the best times to fish. Their elders explain that the stronger spring tides stirred up sediment and obscured fish vision, leaving them more likely to take in lures and resulting in a larger catch.1

私たちの身の回りには、一定の間隔で繰り返す現象が数多くある。潮の満ち引きはその一例で、月の引力に影響され、月の運動に伴ってその大きさも変わる。多くの先住民族は、こうした規則性を物語や文化に取り入れ、農耕・狩猟・漁業の時期を判断するために使ってきた。オーストラリア北端のトレス海峡地域では、潮汐の変化を手掛かりに、漁に適した時期を選んでいた。長老たちによれば、大潮で巻き上げられた堆積物が水を濁らせ、魚の視界を遮るため、魚が仕掛けにかかりやすくなり、漁獲が増えるという。

Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). Dancing with the stars – astronomy and music in the Torres Strait. In Imagining Other Worlds: Explorations in Astronomy and Culture.

In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions.

一定の間隔で同じ値を繰り返す関数を周期関数という。そのグラフには、同じ形が等間隔に繰り返し現れる。各周期の中で対応する位置では、関数は同じ値をとる。したがって、1周期分の値がすべて分かれば、その繰り返しを使って、前後のどの周期の値も求められる。この章では、周期関数のさまざまな例を調べる。